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This announcement provides information on homework, quizzes, and the topics covered in the Statistics for Social Sciences course. It focuses on describing distributions, including their shape, center, and variability.
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Statistics for the Social Sciences Describing Distributions Psychology 340 Spring 2010
Announcements • Homework #1: will accept these on Th (Jan 21) without penalty • Quiz problems • Quiz 1 is now posted, due date extended to Tu, Jan 26th (by 11:00) • Don’t forget Homework 2 is due Tu (Jan 26)
Outline (for week) • Characteristics of Distributions • Finishing up using graphs • Using numbers (center and variability) • Descriptive statistics decision tree • Locating scores: z-scores and other transformations
Distributions • Three basic characteristics are used to describe distributions • Shape • Many different ways to display distribution • Frequency distribution table • Graphs • Center • Variability
Shapes of Frequency Distributions • Unimodal, bimodal, and rectangular
Shapes of Frequency Distributions • Symmetrical and skewed distributions Positively Negatively • Normal and kurtotic distributions
Frequency Graphs • Histogram • Plot the different values against the frequency of each value
Frequency Graphs • Histogram by hand • Step 1: make a frequency distribution table (may use grouped frequency tables) • Step 2: put the values along the bottom, left to right, lowest to highest • Step 3: make a scale of frequencies along left edge • Step 4: make a bar above each value with a height for the frequency of that value
Frequency Graphs • Histogram using SPSS (create one for class height) • Graphs -> Legacy -> histogram • Enter your variable into ‘variable’ • To change interval width, double click the graph to get into the chart editor, and then double click the bottom axis. Click on ‘scale’ and change the intervals to desired widths • Note: you can also get one from the descriptive statistics frequency menu under the ‘charts’ option
Frequency Graphs • Frequency polygon - essentially the same, put uses lines instead of bars
Displaying two variables • Bar graphs • Can be used in a number of ways (including displaying one or more variables) • Best used for categorical variables • Scatterplots • Best used for continuous variables
Bar graphs • Plot a bar graph of men and women in the class • Graphs -> bar • Simple, click define • N-cases (the default) • Enter Gender into Category axis, click ‘okay’
Bar graphs • Plot a bar graph of shoes in closet crossed with men and women • What should we plot? (and why?) • Average number of shoes for each group? • Graphs -> bar • Simple, click define • Other statistic (default is ‘mean’) – enter pairs of shoes • Enter Gender into Category axis, click ‘okay’
Scatterplot • Useful for seeing the relationship between the variables • Graphs -> Legacy Dialogs • Scatter/Dot • Simple Scatter, click ‘define’ • Enter your X & Y variables, click ‘okay’ • Can add a ‘fit line’ in the chart editor • Plot a scatterplot of soda and bottled water drinking
Describing distributions • Distributions are typically described with three properties: • Shape: unimodal, symmetric, skewed, etc. • Center: mean, median, mode • Spread (variability): standard deviation, variance
Describing distributions • Distributions are typically described with three properties: • Shape: unimodal, symmetric, skewed, etc. • Center: mean, median, mode • Spread (variability): standard deviation, variance
Which center when? • Depends on a number of factors, like scale ofmeasurement and shape. • The mean is the most preferred measure and it is closely related to measures of variability • However, there are times when the mean isn’t the appropriate measure.
Which center when? • Use the median if: • The distribution is skewed • The distribution is ‘open-ended’ • (e.g. your top answer on your questionnaire is ‘5 or more’) • Data are on an ordinal scale (rankings) • Use the mode if: • The data are on a nominal scale • If the distribution is multi-modal
Divide by the total number in the population Add up all of the X’s Divide by the total number in the sample The Mean • The most commonly used measure of center • The arithmetic average • Computing the mean • The formula for the population mean is (a parameter): • The formula for the sample mean is (a statistic): • Note: your book uses ‘M’ to denote the mean in formulas
The Mean • Number of shoes: • 5, 7, 5, 5, 5 • 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20,25, 15 • Suppose we want the mean of the entire group? • Can we simply add the two means together and divide by 2? • NO. Why not?
The Weighted Mean • Number of shoes: • 5, 7, 5, 5, 5,30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20,25, 15 • Suppose we want the mean of the entire group? Can we simply add the two means together and divide by 2? • NO. Why not? Need to take into account the number of scores in each mean
Both ways give the same answer The Weighted Mean • Number of shoes: • 5, 7, 5, 5, 5, 30, 11, 12, 20, 14, 12, 15, 8, 6, 8, 10, 15, 25, 6, 35, 20, 20, 20, 25, 15 Let’s check:
The median • The median is the score that divides a distribution exactly in half. Exactly 50% of the individuals in a distribution have scores at or below the median. • Case1: Odd number of scores in the distribution Step1: put the scores in order Step2: find the middle score • Case2: Even number of scores in the distribution Step1: put the scores in order Step2: find the middle two scores Step3: find the arithmetic average of the two middle scores
major mode minor mode The mode • The mode is the score or category that has the greatest frequency. • So look at your frequency table or graph and pick the variable that has the highest frequency. so the mode is 5 so the modes are 2 and 8 Note: if one were bigger than the other it would be called the major mode and the other would be the minor mode
Describing distributions • Distributions are typically described with three properties: • Shape: unimodal, symmetric, skewed, etc. • Center: mean, median, mode • Spread (variability): standard deviation, variance
Variability of a distribution • Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together. • In other words variabilility refers to the degree of “differentness” of the scores in the distribution. • High variability means that the scores differ by a lot • Low variability means that the scores are all similar
m Standard deviation • The standard deviation is the most commonly used measure of variability. • The standard deviation measures how far off all of the scores in the distribution are from the mean of the distribution. • Essentially, the average of the deviations.
-3 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3
-1 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3 4 - 5 = -1
1 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: To get a measure of the deviation we need to subtract the population mean from every individual in our distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1
3 1 2 3 4 5 6 7 8 9 10 μ Computing standard deviation (population) • Step 1: Compute the deviation scores: Subtract the population mean from every score in the distribution. Our population 2, 4, 6, 8 X - μ = deviation scores 2 - 5 = -3 6 - 5 = +1 Notice that if you add up all of the deviations they must equal 0. 4 - 5 = -1 8 - 5 = +3
X - σ = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3 Computing standard deviation (population) • Step 2: Get rid of the negative signs. Square the deviations and add them together to compute the sum of the squared deviations (SS). SS = Σ (X - μ)2 = (-3)2 + (-1)2 + (+1)2 + (+3)2 = 9 + 1 + 1 + 9 = 20
Computing standard deviation (population) • Step 3: Compute the Variance (the average of the squared deviations) • Divide by the number of individuals in the population. variance = σ2 = SS/N • Note: your book uses ‘SD2’ to denote the variance in formulas
standard deviation = σ = Computing standard deviation (population) • Step 4: Compute the standard deviation. Take the square root of the population variance. • Note: your book uses ‘SD’ to denote the standard deviation in formulas
Computing standard deviation (population) • To review: • Step 1: compute deviation scores • Step 2: compute the SS • SS = Σ (X - μ)2 • Step 3: determine the variance • take the average of the squared deviations • divide the SS by the N • Step 4: determine the standard deviation • take the square root of the variance
Computing standard deviation (sample) • The basic procedure is the same. • Step 1: compute deviation scores • Step 2: compute the SS • Step 3: determine the variance • This step is different • Step 4: determine the standard deviation
Our sample 2, 4, 6, 8 1 2 3 4 5 6 7 8 9 10 X - X = deviation scores X Computing standard deviation (sample) • Step 1: Compute the deviation scores • subtract the sample mean from every individual in our distribution. 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3
SS = Σ (X - X)2 2 - 5 = -3 6 - 5 = +1 = (-3)2 + (-1)2 + (+1)2 + (+3)2 4 - 5 = -1 8 - 5 = +3 = 9 + 1 + 1 + 9 = 20 X - X = deviation scores Apart from notational differences the procedure is the same as before Computing standard deviation (sample) • Step 2: Determine the sum of the squared deviations (SS).
3 X X X X 2 1 4 μ Computing standard deviation (sample) • Step 3: Determine the variance Recall: Population variance = σ2 = SS/N The variability of the samples is typically smaller than the population’s variability
Sample variance = s2 Computing standard deviation (sample) • Step 3: Determine the variance Recall: Population variance = σ2 = SS/N The variability of the samples is typically smaller than the population’s variability To correct for this we divide by (n-1) instead of just n
standard deviation = s = Computing standard deviation (sample) • Step 4: Determine the standard deviation
Changes the total and the number of scores, this will change the mean and the standard deviation Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes
X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score
X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score
X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score
X old • All of the scores change by the same constant. Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score
X new • All of the scores change by the same constant. • But so does the mean Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes
X old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes
X old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes
X old • It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Properties of means and standard deviations Mean Standard deviation • Change/add/delete a given score changes changes • Add/subtract a constant to each score changes